“Quasi-stationary distribution”的意思、由来-开放百科全书

In probability a quasi-stationary distribution is a random process that admits one or several absorbing states that are reached almost surely, but is initially distributed such that it can evolve for a long time without reaching it. The most common example is the evolution of a population: the only equilibrium is when there is no one left, but if we model the number of people it is likely to remain stable for a long period of time before it eventually collapses.

Formal definition

We consider a Markov process

taking values in

. There is a measurable set

of absorbing states and

. We denote by

the hitting time of

, also called killing time. We denote by

the family of distributions where

has original condition

. We assume that

is almost surely reached, i.e.

The general definition ^[1] is: a probability measure

is said to be a quasi-stationary distribution (QSD) if for every measurable set

contained in

where

General results

Killing time

From the assumptions above we know that the killing time is finite with probability 1. A stronger result than we can derive is that the killing time is exponentially distributed:^[1]^[2] if

is a QSD then there exists

such that

Existence of a quasi-stationary distribution

Most of the time the question asked is whether a QSD exists or not in a given framework. From the previous results we can derive a condition necessary to this existence.

Let

. A necessary condition for the existence of a QSD is

and we have the equality

Moreover, from the previous paragraph, if

is a QSD then

. As a consequence, if

satisfies

then there can be no QSD

such that

because other wise this would lead to the contradiction

A sufficient condition for a QSD to exist is given considering the transition semigroup

of the process before killing. Then, under the conditions that

is a compact Hausdorff space and that

preserves the set of continuous functions, i.e.

, there exists a QSD.

History

The works of Wright.^[3] on gene frequency in 1931 and Yaglom^[4] on branching processes in 1947 included already the idea of such distributions. The term quasi-stationarity applied to biological systems was then used by Barlett^[5] in 1957, who later coined "quasi-stationary distribution" in.^[6]

Quasi-stationary distributions were also part of the classification of killed processes given by Vere-Jones^[7] in 1962 and their definition for finite state Markov chains was done in 1965 by Darroch and Seneta^[8]

Examples

References

1. ^¹{{Cite book|title=Quasi-Stationary Distributions {{!}} SpringerLink|last=Collet|first=Pierre|last2=Martínez|first2=Servet|last3=Martín|first3=Jaime San|language=en-gb|doi=10.1007/978-3-642-33131-2|series = Probability and its Applications|year = 2013|isbn = 978-3-642-33130-5}}
2. ^{{cite journal|jstor=1427713|title=Existence of Non-Trivial Quasi-Stationary Distributions in the Birth-Death Chain on JSTOR|journal=Advances in Applied Probability|volume=24|issue=4|pages=795–813|language=en|last1=Ferrari|first1=Pablo A.|last2=Martínez|first2=Servet|last3=Picco|first3=Pierre|year=1992}}
3. ^WRIGHT, Sewall. Evolution in Mendelian populations. Genetics, 1931, vol. 16, no 2, pp. 97–159.
4. ^YAGLOM, Akiva M. Certain limit theorems of the theory of branching random processes. In : Doklady Akad. Nauk SSSR (NS). 1947. p. 3.
5. ^BARTLETT, Mi S. On theoretical models for competitive and predatory biological systems. Biometrika, 1957, vol. 44, no 1/2, pp. 27–42.
6. ^BARTLETT, Maurice Stevenson. Stochastic population models; in ecology and epidemiology. 1960.
7. ^{{Cite journal|last=VERE-JONES|first=D.|date=1962-01-01|title=GEOMETRIC ERGODICITY IN DENUMERABLE MARKOV CHAINS|journal=The Quarterly Journal of Mathematics|language=en|volume=13|issue=1|pages=7–28|doi=10.1093/qmath/13.1.7|issn=0033-5606}}
8. ^{{Cite journal|last=Darroch|first=J. N.|last2=Seneta|first2=E.|date=1965|title=On Quasi-Stationary Distributions in Absorbing Discrete-Time Finite Markov Chains|jstor=3211876|journal=Journal of Applied Probability|volume=2|issue=1|pages=88–100|doi=10.2307/3211876}}